**NSF CMMI-1935453: **Declaring War on Boundary Condition: A Control-Oriented
Framework for PDEs

**Project Description:**

The joke goes like this: ``Fusion power is just 20 years away and always will be''. Why 20 years? Because that is how long it takes to build a tokamak fusion reactor. Why always? Because for a tokamak to produce energy, it must control a 10M degree plasma using a magnetic field, a dynamic process governed by six coupled nonlinear Partial-Differential Equations (PDEs) in two spatial variables and no real progress has ever been made on feedback control of such systems. The goal of the project, then, is to make substantial progress on the control of coupled PDEs. Such progress is through foundational advances in mathematics and software infrastructure which will enable the broader PDE controls community to grow. Specifically, by developing a universal, easily-understood and applied computational framework for the control of PDEs, the proposed research will provide the tools needed to allow controls engineers with limited or no PDE experience to design reliable and effective controllers for these systems. Since nuclear fusion reactors have no long-term radioactive waste, and since fuel for these reactors can be extracted from sea water, progress in this area has the potential for creating an unlimited supply of clean energy. Furthermore, this work has the potential to advance several applications beyond nuclear fusion including hypersonic vehicles, traffic management, soft robotics, cavitation, flow control, and vibration suppression.

The goal of the project is to replace Partial Differential Equations (PDEs) with Partial Integral Equations (PIEs). Historically, PDEs are defined by three sometimes contradictory sets of equations and constraints: the PDE itself, which moves the state; the Boundary Conditions (BCs), which implicitly constrain the motion of the state; and the continuity conditions, which couple the BCs to constraints on the motion of the state. By contrast, PIEs combine PDE, BCs, and continuity conditions into a single equation, defined by bounded Partial-Integral (PI) operators, any solution to which satisfies the original PDE and requires no boundary conditions or continuity constraints. PI operators are bounded, form an algebra, and can be parameterized using matrices. Further, positivity of PI operators can be enforced using Linear Matrix Inequalities (LMIs). This means that algorithms developed for control of ODEs using LMIs can be generalized to control of PIEs with relatively little effort. This project will pursue the generalization of several such LMIs to PDEs.

**Project Duration:**

09/01/2019-08/31/2022

**Project Personnel:**

Matthew M. Peet (PI), Arizona State University

Olga Skrowronek (PhD), Arizona State University

**International Collaborators:**

Seip Weiland, Amritam Das, Technical University of Eindhoven, Netherlands

**Software Products:**

DelayTools.PDEs (last updated 03/31/2018)

Based on the Paper:

A New State-Space Representation for Coupled PDEs and Scalable Lyapunov Stability Analysis in the SOS Framework

IEEE Conference on Decision and Control, 2018Introduction: A simplified toolbox for stability analysis of coupled PDEs with generalized boundary conditions.

This toolbox:

1. Requires Matlab 2011a or later.

2. Requires that a working version of SeDuMi be installed.

3. Requires the entire folder be placed in the path along with sub-folders.

4. Replaces any existing version of SOSTOOLS. No previous version should appear in the path.

5. Replaces any existing version of MULTIPOLY. No previous version should appear in the path.

[Delaytools_PDE_stability_vCDC2018_distribution.zip]- DelayTools PDE Analysis Package

**Associated Publications at ASU:**

Student Theses:

Journal Publications:

R. Kamyar and M. M. Peet

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum-of-Squares

Discrete and Continuous Dynamical Systems - Series B. Special Issue on "Constructive and computational methods in Lyapunov and stability theory". Vol. 20, No. 9, pp2383--2417. (Survey Paper).

[arXiv] [.pdf] [.ps]

R. Kamyar, M. M. Peet and Y. Peet

Solving Large-Scale Robust Control Problems by Exploiting the Parallel Structure of Polya's Theorem

IEEE Transactions on Automatic Control, Vol. 58, No. 8, Aug. 2013, pp. 1931-1947.

[arXiv] [.pdf] [.ps]

CDC/ACC Conference Publications:

A. Das and S. Shivakumar and S. Weiland and M. Peet

H-\infty Optimal Estimation for Linear Coupled PDE Systems

IEEE Conference on Decision and Control, 2019.

[arXiv:1809.10308]
[.pdf]
[slides]
[CodeOcean]

S. Shivakumar and A. Das and S. Weiland and M. Peet

Generalized LMI Formulation for Input-Output Analysis of Linear Systems of ODEs Coupled with PDEs

IEEE Conference on Decision and Control, 2019.

[arXiv:1904.10091]
[.pdf]
[slides]
[CodeOcean]